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Voter Model Perturbations and Reaction Diffusion Equations
J. Theodore Cox, Syracuse University, NY, Richard Durrett, Duke University, Durham, NC, and Edwin A. Perkins, University of British Columbia, Vancouver, Canada
A publication of the Société Mathématique de France.
cover
Astérisque
2013; 113 pp; softcover
Number: 349
ISBN-13: 978-2-85629-355-3
List Price: US$52
Member Price: US$41.60
Order Code: AST/349
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The authors consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions \(d \ge 3\). Combining this result with properties of the P.D.E., some methods arising from a low density super-Brownian limit theorem, and a block construction, the authors give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type.

As applications, the authors describe the phase diagrams of four systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin, and (iv) a voter model in which opinion changes are followed by an exponentially distributed latent period during which voters will not change again.

The first application confirms a conjecture of Cox and Perkins, and the second confirms a conjecture of Ohtsuki et al. in the context of certain infinite graphs. An important feature of the authors' general results is that they do not require the process to be attractive.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in voter model perturbations and reaction diffusion equations.

Table of Contents

  • Introduction and statement of results
  • Construction, duality and coupling
  • Proofs of Theorems 1.2 and 1.3
  • Achieving low density
  • Percolation results
  • Existence of stationary distributions
  • Extinction of the process
  • Bibliography
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